Question

In Problems 7-44, follow Steps 1 through 7 on page 228 to analyze the graph of each function.

$f\left(x\right)=2x+\frac{9}{x^3}$

Short answer

The graph is

Step-by-step solution

Given information

Given the function$f\left(x\right)=2x+\frac{9}{x^3}$

Factoring the numerator and denominator

Factoring, we get

$$\begin{gathered} f\left(x\right)=2x+\frac{9}{x^3} \\ f\left(x\right)=\frac{2x^4+9}{x^3} \end{gathered}$$

Calculate the domain

Calculating, we get

$$\begin{gathered} x^3=0 \\ x=0 \end{gathered}$$

The domain is all real numbers$0$

Calculate the x intercept

Setting numerator to zero, we get

$$\begin{gathered} 2x^4+9=0 \\ {x}^4=-\frac{9}{2} \end{gathered}$$

There is no$x,y$intercept

Calculate the vertical asymptote

The zeroes of denominator is the vertical asymptote so$x=0$is the vertical asymptote

Calculating the horizontal asymptote

The degree of numerator is greater than denominator so there is an oblique asymptote

$y=2x$is the oblique asymptote

$$\begin{gathered} f\left(x\right)=\frac{\left(2x^4+9\right)}{x} \\ =2x \\ 2x^4+9=2x^2 \\ 2x^4-2x^2=-9 \end{gathered}$$

It doesn't intersect$f$

Sketching the graph

The graph is